logo AFST
On log K-stability for asymptotically log Fano varieties
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 5, pp. 1013-1024.

The notion of asymptotically log Fano varieties was given by Cheltsov and Rubinstein. We show that, if an asymptotically log Fano variety (X,D) satisfies that D is irreducible and -K X -D is big, then X does not admit Kähler-Einstein edge metrics with angle 2πβ along D for any sufficiently small positive rational number β. This gives an affirmative answer to a conjecture of Cheltsov and Rubinstein.

La notion de variété asymptotiquement log Fano a été proposée par Cheltsov et Rubinstein. Dans ce travail on montre que, si une variété asymptotiquement log Fano (X,D) vérifie que D est irréductible et -K X -D est big, alors X n’admet pas de métrique Kähler-Einstein conique d’angle 2πβ sur D, quelque soit l’angle rationnel positif β suffisamment petit. Ce résultat donne une réponse positive à une conjecture de Cheltsov et Rubinstein.

Published online:
DOI: 10.5802/afst.1520
Kento Fujita 1

1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
@article{AFST_2016_6_25_5_1013_0,
     author = {Kento Fujita},
     title = {On log {K-stability} for asymptotically log {Fano} varieties},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1013--1024},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {5},
     year = {2016},
     doi = {10.5802/afst.1520},
     mrnumber = {3582118},
     zbl = {1375.14140},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1520/}
}
TY  - JOUR
TI  - On log K-stability for asymptotically log Fano varieties
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2016
DA  - 2016///
SP  - 1013
EP  - 1024
VL  - Ser. 6, 25
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1520/
UR  - https://www.ams.org/mathscinet-getitem?mr=3582118
UR  - https://zbmath.org/?q=an%3A1375.14140
UR  - https://doi.org/10.5802/afst.1520
DO  - 10.5802/afst.1520
LA  - en
ID  - AFST_2016_6_25_5_1013_0
ER  - 
%0 Journal Article
%T On log K-stability for asymptotically log Fano varieties
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2016
%P 1013-1024
%V Ser. 6, 25
%N 5
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1520
%R 10.5802/afst.1520
%G en
%F AFST_2016_6_25_5_1013_0
Kento Fujita. On log K-stability for asymptotically log Fano varieties. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 5, pp. 1013-1024. doi : 10.5802/afst.1520. https://afst.centre-mersenne.org/articles/10.5802/afst.1520/

[1] Birkar (C.), Cascini (P.), Hacon (C. D.) and McKernan (J. M).— Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23, no. 2, p. 405-468 (2010). | Article | MR: 2601039 | Zbl: 1210.14019

[2] Berman (R.).— K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics, arXiv:1205.6214; to appear in Invent. Math. | Article | Zbl: 1353.14051

[3] Cheltsov (I. A.) and Rubinstein (Y. A.).— Asymptotically log Fano varieties, Adv. Math. 285, p. 1241-1300 (2015). | Article | MR: 3406526 | Zbl: 1337.14033

[4] Cheltsov (I. A.) and Rubinstein (Y. A.).— On flops and canonical metrics, arXiv:1508.04634. | MR: 3783790 | Zbl: 1393.14041

[5] Fujita (K.).— On K-stability and the volume functions of -Fano varieties, arXiv:1508.04052. | Article

[6] Kaloghiros (A.-S.), Küronya (A.) and Lazić (V.).— Finite generation and geography of models, arXiv:1202.1164; to appear in Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo. | Article | Zbl: 1369.14025

[7] Kollár (J.) and Mori (S.).— Birational geometry of algebraic varieties, Cambridge Tracts in Math, vol.134, Cambridge University Press, Cambridge (1998). | Article

[8] Lazarsfeld (R.).— Positivity in algebraic geometry, I: Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. (3) 48, Springer, Berlin (2004). | Zbl: 1093.14501

[9] Odaka (Y.).— A generalization of the Ross-Thomas slope theory, Osaka. J. Math. 50, no. 1, p. 171-185 (2013). | MR: 3080636 | Zbl: 1328.14073

[10] Odaka (Y.) and Sun (S.).— Testing log K-stability by blowing up formalism, Ann. Fac. Sci. Toulouse Math. 24, no. 3, p. 505-522 (2015). | Article | MR: 3403730 | Zbl: 1326.14096

Cited by Sources: